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href="https://de.wikipedia.org/wiki/Adams-Williamson-Gleichung" title="Adams-Williamson-Gleichung – German" lang="de" hreflang="de" data-title="Adams-Williamson-Gleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Adams-Williamson" title="Ecuación de Adams-Williamson – Spanish" lang="es" hreflang="es" data-title="Ecuación de Adams-Williamson" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_Adams%E2%80%93Williamson" title="Phương trình Adams–Williamson – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình Adams–Williamson" data-language-autonym="Tiếng Việt" 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.sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Earthquakes" title="Category:Earthquakes">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Earthquake" title="Earthquake">Earthquakes</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Earthquake_-_The_Noun_Project.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/80px-Earthquake_-_The_Noun_Project.svg.png" decoding="async" width="80" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/120px-Earthquake_-_The_Noun_Project.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/160px-Earthquake_-_The_Noun_Project.svg.png 2x" data-file-width="512" data-file-height="410" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Types_of_earthquake" title="Types of earthquake">Types</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mainshock" title="Mainshock">Mainshock</a></li> <li><a href="/wiki/Foreshock" title="Foreshock">Foreshock</a></li> <li><a href="/wiki/Aftershock" title="Aftershock">Aftershock</a></li> <li><a href="/wiki/Blind_thrust_earthquake" title="Blind thrust earthquake">Blind thrust</a></li> <li><a href="/wiki/Doublet_earthquake" title="Doublet earthquake">Doublet</a></li> <li><a href="/wiki/Interplate_earthquake" title="Interplate earthquake">Interplate</a></li> <li><a href="/wiki/Intraplate_earthquake" title="Intraplate earthquake">Intraplate</a></li> <li><a href="/wiki/Megathrust_earthquake" title="Megathrust earthquake">Megathrust</a></li> <li><a href="/wiki/Remotely_triggered_earthquakes" title="Remotely triggered earthquakes">Remotely triggered</a></li> <li><a href="/wiki/Slow_earthquake" title="Slow earthquake">Slow</a></li> <li><a href="/wiki/Submarine_earthquake" title="Submarine earthquake">Submarine</a></li> <li><a href="/wiki/Supershear_earthquake" title="Supershear earthquake">Supershear</a></li> <li><a href="/wiki/Tsunami_earthquake" title="Tsunami earthquake">Tsunami</a></li> <li><a href="/wiki/Earthquake_swarm" title="Earthquake swarm">Earthquake swarm</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Causes</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Fault_(geology)" title="Fault (geology)">Fault movement</a></li> <li><a href="/wiki/Volcano_tectonic_earthquake" title="Volcano tectonic earthquake">Volcanism</a></li> <li><a href="/wiki/Induced_seismicity" title="Induced seismicity">Induced seismicity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Characteristics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Epicenter" title="Epicenter">Epicenter</a></li> <li><a href="/wiki/Epicentral_distance" title="Epicentral distance">Epicentral distance</a></li> <li><a href="/wiki/Hypocenter" title="Hypocenter">Hypocenter</a></li> <li><a href="/wiki/Shadow_zone" title="Shadow zone">Shadow zone</a></li> <li><a href="/wiki/Seismic_wave" title="Seismic wave">Seismic waves</a></li> <li><a href="/wiki/P_wave" title="P wave">P wave</a></li> <li><a href="/wiki/S_wave" title="S wave">S wave</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Measurement</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Seismometer" title="Seismometer">Seismometer</a></li> <li><a href="/wiki/Seismic_magnitude_scales" title="Seismic magnitude scales">Seismic magnitude scales</a></li> <li><a href="/wiki/Seismic_intensity_scales" title="Seismic intensity scales">Seismic intensity scales</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Earthquake_prediction" title="Earthquake prediction">Prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em; line-height:1.25em;"><a href="/wiki/Coordinating_Committee_for_Earthquake_Prediction" title="Coordinating Committee for Earthquake Prediction">Coordinating Committee for<br />Earthquake Prediction</a></div></li> <li><a href="/wiki/Earthquake_forecasting" title="Earthquake forecasting">Forecasting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Other topics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Shear_wave_splitting" title="Shear wave splitting">Shear wave splitting</a></li> <li><a class="mw-selflink selflink">Adams–Williamson equation</a></li> <li><a href="/wiki/Flinn%E2%80%93Engdahl_regions" class="mw-redirect" title="Flinn–Engdahl regions">Flinn–Engdahl regions</a></li> <li><a href="/wiki/Earthquake_engineering" title="Earthquake engineering">Earthquake engineering</a></li> <li><a href="/wiki/Seismite" title="Seismite">Seismite</a></li> <li><a href="/wiki/Seismology" title="Seismology">Seismology</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist"> <ul><li><a href="/wiki/Portal:Earth_sciences" title="Portal:Earth sciences">Earth Sciences Portal</a></li></ul> <ul><li><a href="/wiki/Category:Earthquakes" title="Category:Earthquakes">Category</a></li> <li><a href="/wiki/Index_of_geology_articles" title="Index of geology articles">Related topics</a></li></ul></td></tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Earthquakes" title="Template:Earthquakes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Earthquakes" title="Template talk:Earthquakes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Earthquakes" title="Special:EditPage/Template:Earthquakes"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Adams–Williamson equation</b>, named after <a href="/wiki/Leason_H._Adams" class="mw-redirect" title="Leason H. Adams">Leason H. Adams</a> and <a href="/wiki/Erskine_Douglas_Williamson" title="Erskine Douglas Williamson">E. D. Williamson</a>, is an equation used to determine density as a function of radius, more commonly used to determine the relation between the velocities of <a href="/wiki/Seismic_wave" title="Seismic wave">seismic waves</a> and the <a href="/wiki/Density" title="Density">density</a> of the Earth's interior.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Given the average density of rocks at the Earth's surface and profiles of the <a href="/wiki/P-wave" class="mw-redirect" title="P-wave">P-wave</a> and <a href="/wiki/S-wave" class="mw-redirect" title="S-wave">S-wave</a> speeds as function of depth, it can predict how density increases with depth.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It assumes that the compression is <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a> and that the Earth is spherically symmetric, homogeneous, and in <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equilibrium</a>. It can also be applied to spherical shells with that property. It is an important part of models of the Earth's interior such as the <a href="/wiki/Preliminary_reference_Earth_model" title="Preliminary reference Earth model">Preliminary reference Earth model</a> (PREM).<sup id="cite_ref-Poirier_3-0" class="reference"><a href="#cite_note-Poirier-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adams%E2%80%93Williamson_equation&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Williamson and Adams first developed the theory in 1923. They concluded that "It is therefore impossible to explain the high density of the Earth on the basis of compression alone. The dense interior cannot consist of ordinary rocks compressed to a small volume; we must therefore fall back on the only reasonable alternative, namely, the presence of a heavier material, presumably some metal, which, to judge from its abundance in the Earth's crust, in meteorites and in the Sun, is probably iron."<sup id="cite_ref-Poirier_3-1" class="reference"><a href="#cite_note-Poirier-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adams%E2%80%93Williamson_equation&action=edit&section=2" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The two types of seismic body waves are compressional waves (<a href="/wiki/P-waves" class="mw-redirect" title="P-waves">P-waves</a>) and shear waves (<a href="/wiki/S-waves" class="mw-redirect" title="S-waves">S-waves</a>). Both have speeds that are determined by the <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elastic</a> properties of the medium they travel through, in particular the <a href="/wiki/Bulk_modulus" title="Bulk modulus">bulk modulus</a> <i>K</i>, the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> <i>μ</i>, and the <a href="/wiki/Density" title="Density">density</a> <i>ρ</i>. In terms of these parameters, the P-wave speed <i>v</i><sub>p</sub> and the S-wave speed <i>v</i><sub>s</sub> are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{p}&={\sqrt {\frac {K+(4/3)\mu }{\rho }}}\\v_{s}&={\sqrt {\frac {\mu }{\rho }}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>K</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mi>μ</mi> </mrow> <mi>ρ</mi> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>μ</mi> <mi>ρ</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{p}&={\sqrt {\frac {K+(4/3)\mu }{\rho }}}\\v_{s}&={\sqrt {\frac {\mu }{\rho }}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d074f95dbd87309587269640f9391776edf8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:20.801ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}v_{p}&={\sqrt {\frac {K+(4/3)\mu }{\rho }}}\\v_{s}&={\sqrt {\frac {\mu }{\rho }}}.\end{aligned}}}"></span></dd></dl> <p>These two speeds can be combined in a seismic parameter<br /> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =v_{p}^{2}-{\frac {4}{3}}v_{s}^{2}={\frac {K}{\rho }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>K</mi> <mi>ρ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =v_{p}^{2}-{\frac {4}{3}}v_{s}^{2}={\frac {K}{\rho }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8520b878c99e32611fb218300220626a1a4c802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.632ex; height:5.676ex;" alt="{\displaystyle \Phi =v_{p}^{2}-{\frac {4}{3}}v_{s}^{2}={\frac {K}{\rho }}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table><p><br /> </p><p>The definition of the bulk modulus, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=-V{\frac {dP}{dV}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=-V{\frac {dP}{dV}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b20bf11a69cf2ed3bac93eec76f3e8bf4ac239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.246ex; height:5.509ex;" alt="{\displaystyle K=-V{\frac {dP}{dV}},}"></span></dd></dl> <p>is equivalent to </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\rho {\frac {dP}{d\rho }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\rho {\frac {dP}{d\rho }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceafc818d51cc7c2839de353314a6f77a944e6a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.811ex; height:5.843ex;" alt="{\displaystyle K=\rho {\frac {dP}{d\rho }}.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>Suppose a region at a distance <i>r</i> from the Earth's center can be considered a fluid in <a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">hydrostatic equilibrium</a>, it is acted on by gravitational attraction from the part of the Earth that is below it and pressure from the part above it. Also suppose that the compression is <a href="/wiki/Adiabatic" class="mw-redirect" title="Adiabatic">adiabatic</a> (so <a href="/wiki/Thermal_expansion" title="Thermal expansion">thermal expansion</a> does not contribute to density variations). The <a href="/wiki/Pressure" title="Pressure">pressure</a> <i>P</i>(<i>r</i>) varies with <i>r</i> as </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dP}{dr}}=-\rho (r)g(r),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dP}{dr}}=-\rho (r)g(r),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3de7fb60a94b7d24c5bc2ced057f56bb1bd351e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.385ex; height:5.509ex;" alt="{\displaystyle {\frac {dP}{dr}}=-\rho (r)g(r),}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>where <i>g</i>(<i>r</i>) is the <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> at radius <i>r</i>.<sup id="cite_ref-Poirier_3-2" class="reference"><a href="#cite_note-Poirier-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Combining <b><a href="#math_1">1</a></b>,<b><a href="#math_2">2</a></b> and <b><a href="#math_3">3</a></b> gives the Adams–Williamson equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\rho }{dr}}=-{\frac {\rho (r)g(r)}{\Phi (r)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> <mrow> <mi>d</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\rho }{dr}}=-{\frac {\rho (r)g(r)}{\Phi (r)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a5be98164cf26ccd045e25a7141ee22c9b794c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.677ex; height:6.509ex;" alt="{\displaystyle {\frac {d\rho }{dr}}=-{\frac {\rho (r)g(r)}{\Phi (r)}}.}"></span></dd></dl> <p>This equation can be integrated to obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left({\frac {\rho }{\rho _{0}}}\right)=-\int _{r_{0}}^{r}{\frac {g(r)}{\Phi (r)}}dr,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left({\frac {\rho }{\rho _{0}}}\right)=-\int _{r_{0}}^{r}{\frac {g(r)}{\Phi (r)}}dr,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f2d8b6839d6eee879c28f3b3f11f9869e09d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.855ex; height:6.509ex;" alt="{\displaystyle \ln \left({\frac {\rho }{\rho _{0}}}\right)=-\int _{r_{0}}^{r}{\frac {g(r)}{\Phi (r)}}dr,}"></span></dd></dl> <p>where <i>r</i><sub>0</sub> is the radius at the Earth's surface and <i>ρ</i><sub>0</sub> is the density at the surface. Given <i>ρ</i><sub>0</sub> and profiles of the P- and S-wave speeds, the radial dependence of the density can be determined by numerical integration.<sup id="cite_ref-Poirier_3-3" class="reference"><a href="#cite_note-Poirier-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Adams%E2%80%93Williamson_equation&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/C._M._R._Fowler" class="mw-redirect" title="C. M. R. Fowler">C. M. R. Fowler</a> (2005). The Solid Earth: An Introduction to Global Geophysics. Cambridge University Press. pp. 333–. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-89307-7" title="Special:BookSources/978-0-521-89307-7">978-0-521-89307-7</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Eugene F. Milone; William J.F. Wilson (30 January 2014). Solar System Astrophysics: Planetary Atmospheres and the Outer Solar System. Springer Science & Business Media. pp. 494–. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-9090-6" title="Special:BookSources/978-1-4614-9090-6">978-1-4614-9090-6</a>.</span> </li> <li id="cite_note-Poirier-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Poirier_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Poirier_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Poirier_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Poirier_3-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoirier2000" class="citation book cs1">Poirier, Jean-Paul (2000). <i>Introduction to the Physics of the Earth's Interior</i>. Cambridge Topics in Mineral Physics & Chemistry. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66313-X" title="Special:BookSources/0-521-66313-X"><bdi>0-521-66313-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Physics+of+the+Earth%27s+Interior&rft.series=Cambridge+Topics+in+Mineral+Physics+%26+Chemistry&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=0-521-66313-X&rft.aulast=Poirier&rft.aufirst=Jean-Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdams%E2%80%93Williamson+equation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDziewonskiAnderson1981" class="citation journal cs1"><a href="/wiki/Adam_Dziewonski" title="Adam Dziewonski">Dziewonski, A. M.</a>; <a href="/wiki/Don_L._Anderson" title="Don L. Anderson">Anderson, D. L.</a> (1981). "Preliminary reference Earth model". <i><a href="/wiki/Physics_of_the_Earth_and_Planetary_Interiors" title="Physics of the Earth and Planetary Interiors">Physics of the Earth and Planetary Interiors</a></i>. <b>25</b> (4): 297–356. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981PEPI...25..297D">1981PEPI...25..297D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0031-9201%2881%2990046-7">10.1016/0031-9201(81)90046-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+the+Earth+and+Planetary+Interiors&rft.atitle=Preliminary+reference+Earth+model&rft.volume=25&rft.issue=4&rft.pages=297-356&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0031-9201%2881%2990046-7&rft_id=info%3Abibcode%2F1981PEPI...25..297D&rft.aulast=Dziewonski&rft.aufirst=A.+M.&rft.au=Anderson%2C+D.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdams%E2%80%93Williamson+equation" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Adams–Williamson_equation&oldid=1230920900">https://en.wikipedia.org/w/index.php?title=Adams–Williamson_equation&oldid=1230920900</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Structure_of_the_Earth" title="Category:Structure of the Earth">Structure of the Earth</a></li><li><a href="/wiki/Category:Geophysics" title="Category:Geophysics">Geophysics</a></li><li><a href="/wiki/Category:Ordinary_differential_equations" title="Category:Ordinary differential equations">Ordinary differential equations</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 June 2024, at 12:38<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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